Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{x^2 - 64}{x + 8}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{64} = 8$ So we can rewrite the expression as: $n = \dfrac{({x} + {8})({x} {-8})} {x + 8} $ We can divide the numerator and denominator by $(x + 8)$ on condition that $x \neq -8$ Therefore $n = x - 8; x \neq -8$